When I export RAW photos (Nikon D5000 NEF photos converted in DNG format by lightroom) as JPEG, the only way to get the same color as the original photo is by setting the sRGB color profile in the JPEG export options.

Setting ProPhoto or Adobe RGB gives me different results than the original. And when I open one of those RAW photos with Photoshop, it says the RAW photo incorporated profile is ProPhoto RGB. So shouldn't ProPhoto be the best color profile to use? But why can't I export them using ProPhoto RGB or Adobe RGB color space?

2 Answers
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I expect you are looking at the exported JPEG photos with some program which does not properly take the color space into account — it just assumes sRGB, the standard default. So, only your photos exported to match that expectation work. (Even if your screen is calibrated, your applications might not be so smart.)

Keep in mind that Adobe RGB is a color space mainly optimized for print. While monitors are getting better (and yours may qualify as better), most consumer-grade ones can't cover the whole gamut anyway — there are colors that just can't be shown, so they're "clipped" to colors within the monitors' natural/native color gamut.

This isn't part of your question, but:

ProPhoto is a very large color space designed to be a working space, not used for final results. It's very big, and in fact not only exceeds what your monitor can show but contains "imaginary" colors (that's the technical term) which don't correspond to anything in human vision. That's good for a working space, because you don't accidentally clip colors in the midst of curves adjustments and etc., but it's not very useful for a final format. In fact, it's particularly poor for 8-bit-per-channel JPEG images, because there's only 256 possible values for each channel, and when those are spread out so far and "wasted" on imaginary colors, you have less data for real colors, possibly making posterization an issue in gradients.

So, back to the question: you should save your images in a color space based on the intended display. In the current world, that's largely sRGB, unless you're making your own prints or working with a lab that provides printer-specific color profiles for your use. Adobe RGB may also be an option, but your viewing/presentation choices will be more limited (and I think that's what you're running into here).

Photos whose colors are encoded in any space other than sRGB will appear strange (e.g. very flat and desaturated in some cases) when viewed with software that does not understand color management (e.g. web browsers, file preview, etc).

As @mattdm mentioned, sRGB is a catch-all device-dependent color-space that has been designed to give more-or-less ok results on most monitors. When another color space is used, the same RGB value as in a photo in sRGB space will not correspond to the same color, since each space has its own independent color coordinate system.

What is happening is that when you're looking at the photo with non-color-managed software, the step which converts the RGB value into the correct color is not performed, and therefore the colors you are seeing are off.

The sRGB color space is designed to match an average phosphor covered electron cathode ray tube display using display gamma close to 2.2. Most non-wide gamut displays try to emulate that color space because that is de facto color space for all software that is not color management aware. A good operating system would apply automatic from sRGB to display profile color conversion during window rendering unless a piece of software has declared to be color management aware. Unfortunately, no common OS does that. The closest match is Linux + Compiz 0.8.x + CompIcc plugin.
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Mikko RantalainenApr 18 '13 at 6:09